2014 AIME II Problems/Problem 14
14. In △ABC, AB=10, ∠A=30∘, and ∠C=45∘. Let H, D, and M be points on the line BC such that AH⊥BC, ∠BAD=∠CAD, and . Point
is the midpoint of the segment
, and point
is on ray
such that PN⊥BC. Then
, where
and
are relatively prime positive integers. Find
.
http://www.artofproblemsolving.com/Wiki/images/5/59/AOPS_wiki.PNG ( This is the diagram.)
As we can see,
is the midpoint of
and
is the midpoint of
is a
triangle, so ∠HAB=15∘.
is
.
and
are parallel lines so
is
also.
Then if we use those informations we get and
and $AP=AD−PD=2HD−2ND=2HN$ (Error compiling LaTeX. Unknown error_msg) or